Thermal conductivity for phonon scattering by substitutional defects in crystals

Abstract

The thermal conductivity of a harmonic Bravais crystal containing randomly distributed substitutional defects due to impurity-phonon scattering is theoretically investigated using the methods of double-time thermal Green's functions and Kubo formalism considering nondiagonal terms in the heat-current operator as proposed by Hardy. Mass changes as well as force-constant changes between impurity atom and host-lattice atoms are taken into account explicitly. It is shown that the total conductivity can be written as a sum of two contributions, namely, diagonal and nondiagonal contributions. For phonons of small half-width, the former has precisely the same form which is obtained from Boltzmann's transport equation for impurity scattering in the relaxation-time approximation. An analytical expression for the inverse relaxation time due to impurity scattering is obtained in the low-concentration limit of randomly distributed impurities and it shows non-Rayleigh behavior. The non-Rayleigh terms are held responsible for the asymmetric depression in the peak of the thermal-conductivity curves observed experimentally in doped crystals. It is found that the mass change and force-constant change make reinforcing or cancelling contributions to the inverse relaxation time according to whether they are of equal or of opposite signs.